A novel Trefftz method of the inverse Cauchy problem for 3D modified Helmholtz equation

Abstract

We solve an overspecified as well as underspecified inverse Cauchy problems of the modified Helmholtz equation in an arbitrary 3D-bounded domain with a smooth boundary using a multiple/scale/direction Trefftz method (MSDTM), of which the directions are uniformly distributed on , and the scales are determined a priori by the collocation points on boundary to satisfy the specified boundary conditions. The three-dimensional numerical examples confirm the efficiency of the MSDTM. Although under a large noise up to 30%, the solutions of inverse Cauchy problems are quite accurate. More importantly, the present method converges very fast and saves much CPU time.